I have a 10 year old son (I'm English hubby Swiss) he is in the local school. His English and German are the same level - he doesn't mix languages and only ever speaks to me in SG if his friends are around so to include them. (or if he really wants to try and wind me up!)
I'm a language teacher and linguist and so didn't really leave this to chance

From 0-2 I didn't work at all (apart from 1 evening a week) and he heard me all day and everyday. I mixed with other English speaking mums so he could "socialise" in English. We visited family in the UK 4/5 times a year and he built an very powerful bond with his family there (they would come over here 2/3 times too) I think having a strong connection and identity with the "other" country is also really important. My son is Swiss while in Switzerland but the moment we arrive in Manchester he is English (that's what he says)
From 2-4 (pre kindergarten) I worked one morning a week and he went into Swiss day care from 9-14.00 once a week and went to English "playgroup" where I left him once a week for a morning.
Kindergarten age: Until this time his English was better than his SG - but then the SG really kicked in. I balanced this by sending him to English school every Wednesday afternoon (The open door in Basel) where he learned to read, write in English. Now in year 4 primary - he still goes to the Wednesday afternoon English school.
I refuse point blank to communicate with him if he ever speaks SG with me. The only time I budge on that is if his friends are around (to be inclusive)
I still read to him everynight - it's our time together....I'll read around 20 mins and then leave him to continue reading for another 20 (we're on our 4th Harry Potter book)
Strict rule - only English TV before 17.00.
It's worked really well until now. I'm pretty confident if we moved to the UK tomorrow he'd be able to hold his own as a native speaker 10 year old at school.
But it does take some work - and being pretty consistent. I don't really agree that kids will "somehow become bilingual" in a situation where the languages are mixed up and they speak SG and are answered to in English. What tends to happen in such situations is they'll grow up with an excellent understanding of English - but their range of vocabulary and perhaps even accent will be on the level of a very good non-native speaker.
I don't know how much you work - but perhaps if your daughter is spending more than 2 full days a week in Tagi then it might be worth considering limiting it to 2 days and then looking into English speaking childcare for the other days?